In order to have a more flexible distribution on the clustering of the data, in most implementations of the Dirichlet process mixture a prior \(\pi \)  for \(\alpha \)  is specified, leading to a mixing measure that is itself a mixture in the sense of [1]}. Here we show that introducing such a prior has a major impact on the asymptotic behaviour of the number of clusters, as Dirichlet process mixtures can be consistent for the number of clusters.
We provide consistency results under fairly general conditions on \(\pi \)  and for a moderately large class of kernels \(k\) , including uniform and truncated normal distributions.
Following [2]}, we focus on data-generating mixtures with a single component. Our results also extend to the more general case of finite mixtures with multiple components, when a suitable separation assumption between the elements of the mixtures is fulfilled. Crucially, we prove consistency for cases where using a non-random \(\alpha \)  yields inconsistency, thus suggesting that a hyperprior may be beneficial even beyond the cases considered here.
We stress that the framework we study is arguably closer to the way Dirichlet process mixtures are used in practice, compared to holding \(\alpha \)  fixed.
